M Dashti and J C Robinson 2009 Nonlinearity 22 735 doi:10.1088/0951-7715/22/4/003
A simple proof of uniqueness of the particle trajectories for solutions of the Navier–Stokes equations
M Dashti and J C Robinson
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We give a simple proof of the uniqueness of fluid particle trajectories corresponding to (1) the solution of the two-dimensional Navier–Stokes equations with an initial condition that is only square integrable and (2) the local strong solution of the three-dimensional equations with an H1/2-regular initial condition, i.e. with the minimal Sobolev regularity known to guarantee uniqueness. This result was proved by Chemin and Lerner (1995 J. Diff. Eqns 121 314–28) using the Littlewood–Paley theory for the flow in the whole space 
, d ≥ 2. We first show that the solutions of the differential equation 
are unique if u 
Lp(0, T; H(d/2)−1) for some p > 1 and 
. We then prove, using standard energy methods, that the solution of the Navier–Stokes equations with initial condition in H(d/2)−1 satisfies these conditions. This proof is also valid for the more physically relevant case of bounded domains.
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Issue 4 (April 2009)
Received 25 April 2008, in final form 24 January 2009
Published 20 February 2009
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A simple proof of uniqueness of the particle trajectories for solutions of the Navier–Stokes equations
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